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The twelve rotational symmetries of a tetrahedron act on its six edges. Their edge cycle patterns are six fixed edges for the identity, two three-cycles for each of eight rotational symmetries of order three, and two fixed edges plus two transpositions for each of three half-turns. Burnside's lemma therefore counts edge colourings with $q$ labelled colours by $(q^6+8q^2+3q^4)/12$.

The Burgers' equation with constant linear damping has the displayed form. For zero viscosity and $\gamma >0$, characteristics are $u=u_0(a)e^{-\gamma t}$ and $x=a+u_0(a)(1-e^{-\gamma t})/\gamma$. A negative initial slope can make this coordinate map cease to be invertible, producing shock formation by characteristic intersection. The transformation $u=e^{-\gamma t}v$, $\tau =(1-e^{-\gamma t})/\gamma$ converts the inviscid equation to the ordinary inviscid Burgers' equation. Across a thin viscous layer the damping contributes only at lower order to the leading Rankine-Hugoniot condition.

Ventilation of a hospital patient-care space. Air exchange, distribution and source control jointly affect airborne contamination, connecting a building system with infection-control questions.

Ventilation flow divided by the relevant room volume, commonly expressed per hour. It is an exchange-rate measure; incomplete mixing means the same nominal value need not produce identical contaminant removal everywhere.^[ref]

For ${\alpha }_t+[V_s\alpha (1-\alpha ){]}_x=0$ in a finite upward pipe, bubbles move upward at $V_s(1-\alpha )$, but dense-mixture phase-fraction characteristics move downward. Hence an open upper end may be an incoming boundary for the scalar conservation law even though the bubbles themselves leave through it. Atmospheric pressure alone does not prescribe that incoming phase-fraction trace.

The bottom-clearing shock from zero to a uniform initial ${\alpha }_0$ has speed $V_s(1-{\alpha }_0)$. Continuing that constant right state to the outlet gives clearance time $L/[V_s(1-{\alpha }_0)]$, but at ${\alpha }_0>1/2$ it requires a compatible maintained incoming trace. For ${\alpha }_0=3/4$, the shock is $x=V_{s}t/4$, and a right-state characteristic traced backward from it has initial position $3V_{s}t/4$. It reaches the upper boundary at time $4L/(3V_s)$, before the formal clearance time $4L/V_s$. Later evolution therefore depends on the top closure. This boundary issue is distinct from the validity of the local slip and conservation derivation.

A continuous vector field on a closed Euclidean ball whose boundary values have nonnegative outward component has a zero in the ball. Otherwise $T(x)=-Rg(x)/|g(x)|$ is a continuous self-map of the ball. Its fixed point, supplied by the Brouwer fixed-point theorem, lies on the boundary and satisfies $g(x)\cdot x=-R|g(x)|<0$, a contradiction. The argument turns a zero-finding problem into a fixed point problem without any differentiability assumption.

A continuous Euclidean self-map with this growth has a fixed point. Choose $R\ge (2A{)}^{1/(1-\alpha )}$ and $R\ge 2B$. Then $|f(x)|\le R$ whenever $|x|\le R$, so the closed radius-$R$ ball is invariant and Brouwer applies. Existence does not imply convergence of iterating the original map: $x\mapsto 1-x$ on the unit interval has a fixed point but a two-cycle from either endpoint.

Return of breast cancer after treatment, distinguished as local, regional or distant disease.

For offspring $\mathrm{Bin}(k,p)$ with $0<p<1$ and $kp>1$, let $\eta \in (0,1)$ be the branching extinction probability. Conditioning on extinction changes the offspring law to $\mathrm{Bin}(k,\tilde{p})$ with the displayed parameter. Indeed, the probability of $j$ children is multiplied by ${\eta }^{j-1}$, so the new probability generating function is $f(\eta s)/\eta$. Since $f(s)=(1-p+ps{)}^k$ and $f(\eta )=\eta$, this equals $((1-p+p\eta s)/(1-p+p\eta ){)}^k$. Its mean is $k\tilde{p}=f^{\prime }(\eta )<1$, because strict convexity of $f(s)-s$ and its zeros at $\eta$ and one force a negative derivative at the smaller zero. The conditioned process therefore has an exponentially decaying total-progeny tail. Unconditionally that tail instead converges to the survival mass $1-\eta$, with an exponentially small correction. When extinction has probability zero, conditioning on it does not define this law.

If a minimum one-tree is not a tour, some vertex has degree greater than two, since there are equally many edges and vertices. Every tour must omit at least one of its incident selected edges. Partition tours by the first omitted edge in an ordered list, forcing all preceding listed edges and forbidding that edge. These disjoint children cover all tours. Recompute the constrained one-tree lower bound in each child and compare it with the incumbent solution to prune.

If $V$ is uniform on $(0,1)$, the displayed random variable is not standard normal. For $V\le 1/2$ it is nonnegative; for $V>1/2$, its radius is at most $\sqrt{2\log 2}$, so it is always bounded below by $-\sqrt{2\log 2}$. A standard normal distribution has an unbounded lower tail.

The Box-Muller transform instead uses a common radius $\sqrt{-2\log U}$ independent of the angle $2\pi V$ in both Cartesian coordinates. Their polar density is $r{e}^{-r^2/2}/(2\pi )$, and dividing by the coordinate Jacobian $r$ gives the product of two standard normal densities. Changing a logarithm's uniform input changes the distribution, even if the resulting expression superficially resembles that formula.

Bone marrow is tissue within the internal spaces of bones. Hematopoietic regions produce blood cells, while adipose-rich regions have a different composition. Marrow is therefore not simply an empty bone cavity or the mineralized bone matrix itself.

The sphenoid bone is a central skull-base bone with contributions to the orbit and middle cranial region. Its openings and processes relate to multiple nerves, vessels and muscular attachments. That central position makes it important for understanding neighboring skull compartments.

The ethmoid bone contributes to the skull base, nasal cavity and medial orbital walls. Its perforated superior region permits passage of olfactory nerve fibers. One bone therefore participates in several anatomical boundaries and in a sensory pathway.

The lacrimal bone is a small bone in the medial orbital wall associated with the lacrimal drainage region. Its position contributes to the space around the lacrimal sac. It is structurally distinct from the secretory lacrimal gland despite the shared name.

The zygomatic bone forms part of the cheek prominence and lateral orbital boundary. Its connections with neighboring bones help transmit forces through the face. Its external prominence and contribution to the orbit are different aspects of one structural element.

An osteophyte is a bony outgrowth, commonly at a joint margin in a remodeling or degenerative process. Its presence is a structural finding rather than a direct measure of pain or functional impairment. Location and relation to adjacent structures affect its significance.

The mineral-rich residue left after combustion of bone organic material. It contains calcium phosphates and should not automatically be treated as pure tricalcium phosphate.

An epiphysis is the end region of a developing long bone, associated with a secondary ossification center. During growth, the growth-plate region separates it from the shaft-side region. The developmental term should be distinguished from the unrelated historical use for the pineal organ.

The patella is a sesamoid bone within the knee-extension apparatus. It receives the quadriceps tendon proximally and connects through the patellar ligament distally. By changing the geometry of this force-transmission path, it influences the leverage of knee extension.